3) Fundamentals of Vectors

Fundamentals of Vectors

Fundamentals of Vectors

1. Definition of a Vector

A vector is a mathematical object that has both magnitude (size/length) and direction. They allow us to describe relationships and movements in ways scalars cannot.

Real-World Examples:

  • Velocity: Speed with direction (e.g., 60 km/h east).
  • Force: A push or pull applied in a direction.
  • Electric Fields: Represented as vectors at every point in space.

2. Notation of Vectors

Vectors can be represented in different forms:

  • Column Form: \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}
  • Unit Vector Notation: \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}
  • Graphical Representation: Drawn as an arrow in a coordinate plane.

3. Types of Vectors

  • Zero Vector: A vector with no magnitude or direction.
  • Unit Vector: A vector with a magnitude of 1, representing direction only.
  • Equal Vectors: Same magnitude and direction, even if placed differently.
  • Parallel Vectors: Scalar multiples of each other.

4. Vector Operations

  • Addition: Component-wise sum of two vectors.
  • Scalar Multiplication: Scaling a vector’s magnitude.
  • Subtraction: Equivalent to adding a negative vector.

5. Linear Combinations and Span

A vector can be formed as a weighted sum of others:

c1 * v1 + c2 * v2 + ... + cn * vn

In 2D, two non-parallel vectors can generate the entire plane.

6. Linear Independence and Basis Vectors

  • Independent Vectors: None can be expressed as a combination of others.
  • Dependent Vectors: At least one is a redundant combination.
  • Basis Vectors: The minimal set needed to describe a space.

7. Magnitude (Norm) of a Vector

The length of a vector is given by:

|v| = sqrt(v1² + v2² + ... + vn²)

8. Dot Product

Measures similarity between two vectors:

a · b = |a| |b| cos(θ)

9. Cross Product (3D Only)

Produces a perpendicular vector to both inputs:

a × b = determinant of 3x3 matrix

10. Vector Projections

Decomposes one vector along another’s direction:

proj_b a = (a · b) / |b|² * b

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